br At appropriate P T conditions a

At appropriate P-T conditions, a complete series of Rw solid solutions exists in the system Mg2SiO4–Fe2SiO4. Their volume–composition relationship was demonstrated to obey the Vegard\'s law (ideal mixing) by some experimental studies (Akimoto and Fujisawa, 1968; Ringwood and Major, 1970; Akimoto, 1972; Suito, 1972; Akaogi et al., 1989). In contrast, the experimental data on the Fe–Mg partitioning between Rw and coexisting phases indicated a non-ideal mixing behavior for the Rw solid solutions (Frost, 2003a, 2003b). As a rule of thumb, a relatively fast crystal structure adjustment and a relatively slow composition modification due to sluggish chemical diffusion should be anticipated to take place during sample-quenching at subsolidus condition, so that a non-ideal solid solution behavior is more likely for the Rw solid solutions. We have collected 40 unit-cell volume data for the anhydrous ringwoodites (Table 3), and plotted them in Fig. 3(a). In synthase to the observation once made by all early volume-composition studies (Akimoto and Fujisawa, 1968; Ringwood and Major, 1970; Akimoto, 1972; Suito, 1972; Akaogi et al., 1989), a small S-shaped deviation from the Vegard\'s law is vaguely demonstrated (thin broken curve). Although it is observed in the ringwoodites for the first time, this type of non-ideal volume-mixing behavior, with negative deviation from linearity near the small-volume end-member (Mg2SiO4-Rw in our case) and positive deviation near the large-volume end-member (Fe2SiO4-Rw in our case), is a general phenomenon rather than an exception for the silicate solid solutions (Newton and Wood, 1980). This implies that the exact mixing behavior in this system still needs to be experimentally characterized in order to accurately thermodynamically describe the Rw. To achieve this goal, several factors such as the water content, the cation order-disorder state and the iron oxidation state have to be simultaneously evaluated along with the XFe parameter. In addition, in situ measurements should be conducted since the cation order-disorder state may not be fully preserved during sample-quenching (O\'Neill et al., 2003). Since the deviation from the Vegard\'s law is so small and can not be well constrained by the available experimental data, anyhow, we presently choose the following weighted linear equation to describe the volume-composition data in the system Mg2SiO4–Fe2SiO4: Using our empirical model (equation (3)), we next calculate the isothermal bulk moduli of these ringwoodites, and show the result in Fig. 3(b). The resulting equation relating the isothermal bulk modulus and composition of the ringwoodites is: Compared to the estimates made in Weidner et al. (1984), Sinogeikin et al. (1998), Higo et al. (2006) and Liu et al. (2008a), our result is substantially smaller, by at least 53% (Table 4). Our result thus supports the experimental observation made by Mao et al. (1969) and Nestola et al. (2010), a negligible effect of the Mg/Fe substitution on the bulk modulus of the Rw solid solutions. The much larger composition dependence of the isothermal bulk modulus of the Rw observed in other studies was perhaps caused by some flaws in the employed experimental methods and/or some fluctuations of some microproperties in the investigated samples, as discussed earlier.
Despite all the potential experimental uncertainties mentioned earlier, it is still highly desirable to directly measure the elastic wave velocities of the Rw solid solutions in the system Mg2SiO4–Fe2SiO4 by using the ultrasonic method (Li et al., 1996, 1998) or Brillouin scattering technique (Weidner et al., 1984; Sinogeikin et al., 1997, 1998). Such measurements were once made either at high temperature and ambient pressure, or at high pressure and ambient temperature, and inevitably led to some extra uncertainties in the sound velocities at simultaneously high-P and high-T conditions, due to the long distance extrapolation in the P-T space (Li and Liebermann, 2007). Recently, direct sound velocity measurements become possible at P-T conditions generally matching those in the lower part of the MTZ (Higo et al., 2008; Irifune et al., 2008), but the available experimental facilities are still very limited.